Higher harmonics in planar Hall effect induced by cluster magnetic multipoles

Antiferromagnetic (AFM) materials are attracting tremendous attention due to their spintronic applications and associated novel topological phenomena. However, detecting and identifying the spin configurations in AFM materials are quite challenging due to the absence of net magnetization. Herein, we report the practicality of utilizing the planar Hall effect (PHE) to detect and distinguish “cluster magnetic multipoles” in AFM Nd2Ir2O7 (NIO-227) fully strained films. By imposing compressive strain on the spin structure of NIO-227, we artificially induced cluster magnetic multipoles, namely dipoles and A2- and T1-octupoles. Importantly, under magnetic field rotation, each magnetic multipole exhibits distinctive harmonics of the PHE oscillation. Moreover, the planar Hall conductivity has a nonlinear magnetic field dependence, which can be attributed to the magnetic response of the cluster magnetic octupoles. Our work provides a strategy for identifying cluster magnetic multipoles in AFM systems and would promote octupole-based AFM spintronics.

In general, I think the work is carefully done and in principle worthy of publication in Nature Comm. However, the manuscript is not publishable in its present form, and the authors need to respond to my comments and revise the manuscript accordingly. 1. The theoretical analysis (Sec. III of the SI and also "Self-consistent mean-field Hubbard model calculations" in the Methods section leaves a lot out. The Hubbard Hamiltonian Eq. (7) (Eq. (3) in the SI) has a bunch of parameters that are not defined (the vectors d_ij, D_ij, and R_ij) nor are values for them given. Also, the self-consistent calculations indicate that A_2 order exists even in the absence of strain. Does that not mean that a sin(4\phi) term should show up in the magnetoresistance even in the absence of strain? Also, how are the order parameters A_2 and T_1 defined in terms of the operators of the Hubbard Hamiltonian? I imagine that the results presented in Fig.s4 are expectation values of those operators. Is that correct? 2. Is the temperature dependence the same for the dipolar and octupolar orders? That is, does octupolar order set in at the Neel temperature or is there another temperature scale for them? The authors indicate that the contributions to the Hall conductivity are visible below 15 K. Is that some particular scale or just what the authors observe for no particular reason? Also, why are the octupolar contributions to the Hall resistivity so small? Is there a way to understand that from the Hubbard Hamiltonian? 3. Stylistically, there is a lot of redundancy between the SI and the manuscript (in particular the theoretical discussions in the Methods section and the SI). The authors may want to think about revising them both so that the manuscript reads as a self-contained manuscript and the SI fill in details. 4. The English, although pretty good, has some some missing articles and there are some awkward passages, such as the top of p. 4"...cluster magnetic octupole (CMO) without magnetization that has the same symmetry as belongs to...."; I think one should say "dipolar order" instead of "dipole order" (top of p. 8). There are other examples -I urge the authors to carefully proofread a revised manuscript and get assistance from a native-English speaker. 5. The authors mention relations between breaking time-reversal symmetry and the appearance of non-trivial topology. In my understanding, a system that respects the combined symmetry operations of time reversal symmetry and any unitary operator cannot have a non-trivial topology. My point is that the text in the manuscript can be construed to say that breaking only TRS is enough to give rise to non-trivial topology, which it is not; the authors should be a little bit more careful in statements relating symmetries and non-trivial topology. 6. I think there is some interesting work by Broholm and Tchernyshyov and co-workers on Mn3Ge (Physical Review B, 102, 054403, 2020 and references therein) that perhaps indicate similar physics, such as the emergence of a magnetization component perpendicular to an applied field. I think it would be interesting if the authors comment on this class of kagome antiferromagnets (beyond the discussions in Refs. 26 and 28).
Reviewer #3: Remarks to the Author: Song et al. describes experimental study on planar Hall effect (PHE) found in antiferromagnetic conductor Nd2Ir2O7, well known for its magnetic structure that can host topological phases. The key finding is that the PHE shows higher harmonics (4th, 6th) in addition to standard second order term as a function of magnetic-field angle. The authors attribute the origin of higher harmonics in PHE to the cluster magnetic octupole that arises in the strained Nd2Ir2O7 films.
The higher order effect in PHE has been theoretically discussed in recent literature, but experimental data has been rather scarce so far. The present data will add new experimental evidence on the higher harmonics PHE. Also, I believe that the use of PHE to potentially probe octupole order in magnetic materials is novel and interesting, and likely would trigger related efforts to unveil magnetic orders by PHE difficult to probe by other means. All this being said, I believe that the manuscript requires some clarifications and serious revisions which are detailed below:

1.The octupole order
What is the independent experimental evidence of octupole order besides PHE? One peripheral signature for the cluster magnetic octupole seems to be the lack of net magnetization. However, there are no direct magnetization measurements performed on these films. Authors should include magnetic measurements performed, e.g., by SQUID. Since anomalous hall effect clearly shows hysteric behavior normally indicative of net magnetization, it seems important to have direct data on the lack of long-range magnetic order. Even so, I am left wondered what the real experimental evidence of "cluster magnetic octupole (CM)" is in the films reported here at this point.

2.Intro
Introduction contains many general aspects on magnetism and spin ordering, but lacks introduction on how the PHE could help identifying those phases, which readers would be more interested. Authors should consider bringing PHE into a spotlight much earlier in the text to improve readability. In the current version, there are too many detailed discussion on general magnetic order, symmetry breaking, induced topology etc., which are not urgently relevant to understand the authors' work. Besides, first two paragraphs of "Results" section are in fact not results, but again the introduction to magnetic order, which need to be included in the Introduction section. I suggest authors to sharply edit their introduction to highlight core insights that lead to their work, and especially to include the mechanisms how and why PHE can probe the octupole order at least appearing in the second or third paragraph of the introduction.

Author's Response:
We greatly acknowledge the reviewer for the positive opinion. We have answered all questions from the reviewer below.
1. To my understanding, the AIAO spin texture is formed when compressive strain is applied, and the free energy is composed of contributions from magnetic multipoles. I am curious about the magnetic anisotropy which is seemed to have been obviated in this model. Would the anisotropy be still negligible under strained conditions? Would it affect the analysis of the multipoles? Perhaps a scan of an out-of-plane angel would help to illustrate more about this point.

Response:
We appreciate the constructive question raised by the reviewer.
Before we go into a detailed discussion about the magnetic anisotropy in pyrochlore iridates, we would like to clarify the emergence of AIAO spin texture in pyrochlore iridates.

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The magnetic anisotropy in this system comes from Dzyaloshinskii-Moriya (DM) interaction and single-ion anisotropy. The Ir spins have DM interaction which gives rise to magnetic anisotropy and AIAO spin texture as mentioned above. The Nd spins has strong single-ion anisotropy, because of high spin-9/2. Note that the energy of single-ion anisotropy is proportional to the spin operator square S z 2 , and single-ion anisotropy for spin-1/2 is absent (S z 2 = 1). The single-ion anisotropy of Nd spins affects Ir spin configuration and the transport by Ir electrons through a strong f-d exchange coupling. We have considered the DM interaction in the Hubbard model in Supplementary Materials. The hopping includes the spin-orbit coupling, which acts as the DM interaction in the large U limit.
Response: We thank the reviewer for this important question. Following the suggestion, we performed the angle-dependent magnetoresistance measurement. Fig. R3 is the anisotropic magnetoconductance (AMC) below 30 K. We observed the development of complex features in AMC below 15 K. To understand this feature of AMC, we theoretically calculated the AMC induced by a dipole, A2 octupole, and T1 octupole.
We use Onsager's relation for longitudinal conductivity from cluster multipoles.
In general, I think the work is carefully done and in principle worthy of publication in Nature Comm. However, the manuscript is not publishable in its present form, and the authors need to respond to my comments and revise the manuscript accordingly.

Author's Response:
We thank the reviewer very much for the recommendation. We have answered all the questions raised and revised the manuscript carefully.
1. The theoretical analysis (Sec. III of the SI and also "Self-consistent mean-field Hubbard model calculations" in the Methods section leaves a lot out. The Hubbard Hamiltonian Eq. (7) (Eq. (3)  We explain the relationship between octupolar orders and the Hall conductivity. First, we here address the relation of octupolar orders to anomalous Hall conductivity. We recall that there are two kinds of octupolar orders in the strained Nd2Ir2O7 thin films, A2 and T1-octupoles.
There are several differences between A2 and T1-octupoles. First, A2-octupole is a scalar order parameter, while T1-octupole is a vector one. Second, the magnetic point group for A2-octupole is −4′3m′, while that for T1-octupole is −42′m′. As for why the octupolar contributions to the planar Hall conductivity are small. We calculated the magnetization induced by strain and magnetic field in the octupolar system Nd2Ir2O7 (Fig. R5). We recall that the planar Hall conductivity in Nd2Ir2O7 is generated by the orthogonal magnetization that results from the coupling between the octopolar orders (higherrank multipoles) and the magnetic field. Since orthogonal magnetization is the coupling between the higher rank multipoles and magnetic field, it should have small values. As expected (Fig. R5b), the magnetization (orthogonal magnetization) induced by the magnetic field in the strained Nd2Ir2O7 is very small (at the scale of 10 -3 μB/atom). Hence, the octupolar contributions to the planar Hall conductivity are small. We added some clarification in the revised main manuscript and Supplementary Materials, which is highlighted on page 9, lines 9-15 in the revised main manuscript, and pages 5-6 and 19-20 in Supplementary Materials.  3. Stylistically, there is a lot of redundancy between the SI and the manuscript (in particular the theoretical discussions in the Methods section and the SI). The authors may want to think about revising them both so that the manuscript reads as a self-contained manuscript and the SI fill in details.

Response:
We thank the reviewer for pointing this out. We have removed all redundant details in the Methods section. Especially, we found that the theoretical calculation of orthogonal magnetization and PHE from cluster magnetic octupoles in Methods and Supplementary materials are similar. Therefore, we decide to remove the theoretical description of them in the Methods section. Please see the Method section in the revised manuscript.

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4. The English, although pretty good, has some missing articles and there are some awkward passages, such as the top of p. 4"...cluster magnetic octupole (CMO) without magnetization that has the same symmetry as belongs to...."; I think one should say "dipolar order" instead of "dipole order" (top of p. 8). There are other examples -I urge the authors to carefully proofread a revised manuscript and get assistance from a native-English speaker.

Response:
We thank the reviewer for pointing this out. The awkward passage regarding the symmetry of CMO is corrected. Also, we have changed all of the dictation regarding magnetic ordering to "dipolar" and "octupolar" orderings. Furthermore, we also proofread the manuscript by both native-English speaker and Springer Nature English Services.  conductor Nd2Ir2O7, well known for its magnetic structure that can host topological phases.
The key finding is that the PHE shows higher harmonics (4th, 6th) in addition to standard second order term as a function of magnetic-field angle. The authors attribute the origin of higher harmonics in PHE to the cluster magnetic octupole that arises in the strained Nd2Ir2O7 films.
The higher order effect in PHE has been theoretically discussed in recent literature, but experimental data has been rather scarce so far. The present data will add new experimental evidence on the higher harmonics PHE. Also, I believe that the use of PHE to potentially probe octupole order in magnetic materials is novel and interesting, and likely would trigger related efforts to unveil magnetic orders by PHE difficult to probe by other means. All this being said, I believe that the manuscript requires some clarifications and serious revisions which are detailed below:

Author's Response:
We thank the reviewer very much for the positive evaluation. We have answered all the questions raised and revised the manuscript carefully.

1.The octupole order
What is the independent experimental evidence of octupole order besides PHE?
One peripheral signature for the cluster magnetic octupole seems to be the lack of net magnetization. However, there are no direct magnetization measurements performed on these films. Authors should include magnetic measurements performed, e.g., by SQUID. Since anomalous hall effect clearly shows hysteretic behavior normally indicative of net 25 magnetization, it seems important to have direct data on the lack of long-range magnetic order.
Even so, I am left wondered what the real experimental evidence of "cluster magnetic octupole (CM)" is in the films reported here at this point.

Response:
We appreciate the reviewer very much for this important comment. We agree with the reviewer that peripheral evidence of cluster octupole is the lack of magnetization.
Following the suggestion, we performed the magnetization measurement. As expected, the magnetization of strained Nd2Ir2O7 film at 2 K exhibits a lack of magnetization. However, the finite AHE at 2K is observed (Fig. R6).  (2016)]. Moreover, the magnetic spin ordering in thin films cannot be detected by the neutron scattering technique since the film is too thin (the mass of the film is not enough for the neutron scattering). In contrast, the planar Hall effect can be performed in thin film samples. Importantly, under a magnetic field, each cluster multipoles exhibits distinctive harmonics of PHE oscillation in our Nd2Ir2O7 film. Therefore, we believe that the PHE can be the practical way to distinguish and identify antiferromagnetic cluster octupoles.
To clarify the understanding of our findings, we added comments and magnetization data in the revised main manuscript and Supplementary Materials. Please see page 6, lines 15-16 in the revised main manuscript and Fig. s1c in the Supplementary Materials.

2.Intro
Introduction contains many general aspects on magnetism and spin ordering, but lacks introduction on how the PHE could help identifying those phases, which readers would be more interested. Authors should consider bringing PHE into a spotlight much earlier in the text to improve readability. In the current version, there are too many detailed discussions on general magnetic order, symmetry breaking, induced topology etc., which are not urgently relevant to understand the authors' work. Besides, first two paragraphs of "Results" section are in fact not results, but again the introduction to magnetic order, which need to be included in the Introduction section. I suggest authors to sharply edit their introduction to highlight core insights that lead to their work, and especially to include the mechanisms how and why PHE can probe the octupole order at least appearing in the second or third paragraph of the introduction.

Response:
We thank the reviewer for this constructive comment. We have revised the introduction of the manuscript with the constructive suggestion given by the reviewer. The discussions about antiferromagnetic order, spintronics, and topology are shortened and the new paragraph about the planar Hall effect to give spotlight is added to the third paragraph of the introduction. The added paragraph is following: "The planar Hall effect (PHE) has been considered as a method of probing the physical properties of materials such as magnetism and topology. The PHE corresponds to the development of a Hall voltage when electric and magnetic fields are coplanar, which is different from the usual Hall effect where they are perpendicular to each other. Initially, the PHE was observed in ferromagnetic systems, detecting the anisotropic magnetization of ferromagnetic materials. Additionally, the PHE has recently been in the spotlight due to its role in detecting topological characteristics such as the chiral anomaly arising from Weyl points in Weyl semimetals [36][37][38][39] . The associated PHE in both ferromagnets and Weyl semimetals exhibit sin (2or second harmonic PHE oscillations. Recently, higher harmonics PHE oscillations beyond second have been reported in topological systems [41][42][43][44] , which indicates that the PHE can be a practical tool to probe additional hidden parameters in materials such as CMOs in a system."